 
Summary: Optimal ExpectedCase Planar Point Location
Sunil Arya
Theocharis Malamatos
David M. Mount§
Ka Chun Wong
November 20, 2006
Abstract
Point location is the problem of preprocessing a planar polygonal subdivision S of size n
into a data structure in order to determine efficiently the cell of the subdivision that contains a
given query point. We consider this problem from the perspective of expected query time. We
are given the probabilities pz that the query point lies within each cell z S. The entropy H of
the resulting discrete probability distribution is the dominant term in the lower bound on the
expectedcase query time. We show that it is possible to achieve query time H + O(
H + 1)
with space O(n), which is optimal up to lower order terms in the query time. We extend this
result to subdivisions with convex cells, assuming a uniform query distribution within each cell.
In order to achieve space efficiency, we introduce the concept of entropypreserving cuttings.
Preliminary results appeared in the papers "Nearly Optimal ExpectedCase Planar Point Location" in Proc. 41st
